Optimal stopping with a fair die (3-roll horizon)

You observe outcomes of fair six-sided die rolls (faces 1–6) and may stop after any roll to take the shown value in dollars. If you have not stopped before the final allowed roll, you must take the last roll’s value. Use backward induction to find the optimal policy and expected payoff for each game.

Assumptions:

• The die is fair (each face has probability 1/6).
• You observe each roll before deciding whether to stop.
• Where a fee applies, it is paid immediately before taking the next roll (only if you choose to continue).

Game A (no costs)

• Up to three rolls; no fees to continue.
• Find the optimal stopping rule at each roll and the exact expected payoff.

Game B (optional continuation cost)

• Up to three rolls; if you choose to continue after roll 1 or roll 2, you must pay $1 before the next roll (each continuation costs$ 1).
• Find the optimal stopping thresholds (by face value) and the exact expected payoff.

Game C (per-roll access fee)

• Up to three rolls; the second and third rolls each require paying $1 to roll (i.e., if you continue from roll 1 you pay$ 1 to access roll 2, and if you continue from roll 2 you pay $1 to access roll 3).
• Find the optimal policy and expected payoff.

Bonus (generalization)

• Generalize Game B to an n-sided die with faces {1, 2, …, n} and a continuation cost c each time you choose to roll again (up to three rolls total). Derive the stopping thresholds at roll 1 and roll 2 as functions of n and c.